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KatieReddingerFinalThesis.pdf (467.95 KB)

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Numerical Stability & Numerical Smoothness of Ordinary Differential Equations

Reddinger, Kaitlin Sue
http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1431597407

Abstract Details

2015, Master of Arts (MA), Bowling Green State University, Mathematics.
Although numerically stable algorithms can be traced back to the Babylonian period, it is believed that the study of numerical methods for ordinary differential equations was not rigorously developed until the 1700s. Since then the field has expanded - first with Leonhard Euler’s method and then with the works of Augustin Cauchy, Carl Runge and Germund Dahlquist. Now applications involving numerical methods can be found in a myriad of subjects. With several centuries worth of diligent study devoted to the crafting of well-conditioned problems, it is surprising that one issue in particular - numerical stability - continues to cloud the analysis and implementation of numerical approximation. According to professor Paul Glendinning from the University of Cambridge, “The stability of solutions of differential equations can be a very difficult property to pin down. Rigorous mathematical definitions are often too prescriptive and it is not always clear which properties of solutions or equations are most important in the context of any particular problem. In practice, different definitions are used (or defined) according to the problem being considered. The effect of this confusion is that there are more than 57 varieties of stability to choose from” [10]. Although this quote is primarily in reference to nonlinear problems, it can most certainly be applied to the topic of numerical stability in general. We will investigate three of the main issues surrounding numerical stability in the error analysis and show that numerical smoothing should have been the right concept in delivering better error estimations. Therefore, the materials on numerical stability in textbooks and classrooms should be replaced by numerical smoothness.
Tong Sun, Dr. (Advisor)
So-Hsiang Chou, Dr. (Committee Member)
Kimberly Rogers, Dr. (Committee Member)
52 p.
Applied Mathematics; Mathematics Education
numerical stability; numerical smoothing; error splitting; error analysis

Recommended Citations

Citations

  • Reddinger, K. S. (2015). Numerical Stability & Numerical Smoothness of Ordinary Differential Equations [Master's thesis, Bowling Green State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1431597407

    APA Style (7th edition)

  • Reddinger, Kaitlin. Numerical Stability & Numerical Smoothness of Ordinary Differential Equations. 2015. Bowling Green State University, Master's thesis. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1431597407.

    MLA Style (8th edition)

  • Reddinger, Kaitlin. "Numerical Stability & Numerical Smoothness of Ordinary Differential Equations." Master's thesis, Bowling Green State University, 2015. http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1431597407

    Chicago Manual of Style (17th edition)

bgsu1431597407
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© 2015, all rights reserved.
This open access ETD is published by Bowling Green State University and OhioLINK.